Abstract
We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d) time and O(n d−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aelst, S.V., Rousseeuw, P.J.: Robustness of deepest regression. J. Multivar. Anal. 73, 82–106 (2000)
Aelst, S.V., Rousseeuw, P.J., Hubert, M., Struyf, A.: The deepest regression method. J. Multivar. Anal. 81(1), 138–166 (2002)
Amenta, N., Bern, M., Eppstein, D., Teng, S.-H.: Regression depth and center points. Discrete Comput. Geom. 23, 305–323 (2000)
Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proc. 43rd Symposium on Foundations of Computer Science (FOCS), pp. 617–626 (2002)
Brönnimann, H., Chazelle, B.: Optimal slope selection via cuttings. Comput. Geom. Theory Appl. 10(1), 23–29 (1998)
Chan, T.M.: Dynamic planar convex hull operations in near-logarithmic amortized time. J. ACM 48(1), 1–12 (2001)
Chan, T.M.: An optimal randomized algorithm for maximum turkey depth. In: Proc. 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 430–436 (2004)
Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(2), 145–158 (1993)
Chazelle, B., Preparata, F.P.: Halfspace range search: An algorithmic application of k-sets. Discrete Comput. Geom. 1, 83–93 (1986)
Cole, R., Salowe, J., Steiger, W., Szemerédi, E.: An optimal-time algorithm for slope selection. SIAM J. Comput. 18(4), 792–810 (1989)
Dey, T.K.: Improved bounds on planar k-sets and related problems. Discrete Comput. Geom. 19, 373–382 (1998)
Dillencourt, M.B., Mount, D.M., Netanyahu, N.S.: A randomized algorithm for slope selection. Int. J. Comput. Geom. Appl. 2, 1–27 (1992)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, New York (1987)
Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput 15, 341–363 (1986)
Edelsbrunner, H., Seidel, R., Sharir, M.: On the zone theorem for hyperplane arrangements. SIAM J. Comput. 22(2), 418–429 (1993)
Eu, D., Guévremont, E., Toussaint, G.T.: On envelopes of arrangements of lines. J. Algorithms 21, 111–148 (1996)
Guibas, L.J., Overmars, M.H., Robert, J.-M.: The exact fitting problem for points. Comput. Geom. Theory Appl. 6, 215–230 (1996)
Hubert, M., Rousseeuw, P.J.: The catline for deep regression. J. Multivar. Anal. 66, 270–296 (1998)
Katz, M.J., Sharir, M.: Optimal slope selection via expanders. Inf. Process. Lett. 47, 115–122 (1993)
Keil, M.: A simple algorithm for determining the envelope of a set of lines. Inf. Process. Lett. 39, 121–124 (1991)
Langerman, S., Steiger, W.: The complexity of hyperplane depth in the plane. Discrete Comput. Geom. 30(2), 299–309 (2003)
Matoušek, J.: Randomized optimal algorithm for slope selection. Inf. Process. Lett. 39, 183–187 (1991)
Peck, G.W.: On k-sets in the plane. Discrete Math. 56, 73–74 (1985)
Rousseeuw, P.J., Aelst, S.V., Hubert, M.: Regression depth: Rejoinder. J. Am. Stat. Assoc. 94, 419–433 (1999)
Rousseeuw, P.J., Hubert, M.: Depth in an arrangement of hyperplanes. Discrete Comput. Geom. 22, 167–176 (1999)
Rousseeuw, P.J., Hubert, M.: Regression depth. J. Am. Stat. Assoc. 94, 388–402 (1999)
Rousseeuw, P.J., Ruts, I.: Constructing the bivariate Tukey median. Stat. Sin. 8, 827–839 (1998)
Rousseeuw, P.J., Struyf, A.: Computing location depth and regression depth in higher dimensions. Stat. Comput. 8, 193–203 (1998)
Rousseeuw, P.J., Struyf, A.: Computation of robust statistics: depth, median, and related measures. In: Goodman, J.E., O’Rourke, J. (eds.) The Handbook of Discrete and Computational Geometry, 2nd edn., pp. 1279–1292. Chapman & Hall/CRC, Boca Raton (2004)
Ruts, I., Rousseeuw, P.J.: Computing depth contours of bivariate point clouds. Comput. Stat. Data Anal. 23, 153–168 (1996)
Sharir, M.: k-sets and random hulls. Combinatorica 13, 483–495 (1993)
Struyf, A., Rousseeuw, P.J.: Halfspace depth and regression depth characterize the empirical distribution. J. Multivar. Anal. 69, 135–153 (1999)
Struyf, A., Rousseeuw, P.J.: High-dimensional computation of the deepest location. Comput. Stat. Data Anal. 34, 415–426 (2000)
Tóth, G.: Point sets with many k-sets. Discrete Comput. Geom. 26, 187–194 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in the proceedings of the 15th Annual ACM Symposium on Computational Geometry (1999)
M. van Kreveld partially funded by the Netherlands Organization for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.065.503.
J.S.B. Mitchell’s research largely conducted while the author was a Fulbright Research Scholar at Tel Aviv University. The author is partially supported by NSF (CCR-9504192, CCR-9732220), Boeing, Bridgeport Machines, Sandia, Seagull Technology, and Sun Microsystems.
M. Sharir supported by NSF Grants CCR-97-32101 and CCR-94-24398, by grants from the U.S.–Israeli Binational Science Foundation, the G.I.F., the German–Israeli Foundation for Scientific Research and Development, and the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University.
J. Snoeyink supported in part by grants from NSERC, the Killam Foundation, and CIES while at the University of British Columbia.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
van Kreveld, M., Mitchell, J.S.B., Rousseeuw, P. et al. Efficient Algorithms for Maximum Regression Depth. Discrete Comput Geom 39, 656–677 (2008). https://doi.org/10.1007/s00454-007-9046-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-007-9046-6